Transcript Beale (Presentation)

FUEL CELL STACK MODEL BASED ON
MULTIPLIPLY SHARED SPACE METHOD
Steven B. Beale* and Sergei V. Zhubrin**
* National Research Council, Ottawa, Canada
** Flowsolve Ltd, London, United Kingdom
Contents
Introduction to distributed resistance analogy
Introduction to solid oxide fuel cells
Governing equations
Cell and stack geometries considered
Comparison of present model with detailed calculations
Discussion of results
Conclusions and future work
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DISTRIBUTED RESISTANCE
ANALOGY
Problem: Not possible to make grid fine enough to capture
flow around all tubes.
Solution: Prescribe distributed resistance, F, and heat transfer
coefficients, a, and solve for superficial flow around baffles.
Values of f, and Nu, obtained from experiments, analysis, or
detailed numerical simulations.
3
DISTRIBUTED RESISTANCE
ANALOGY
Original scheme developed by Patankar and Spalding (1972)
only shell-side flow solved for. Key concept: replace diffusion
terms with prescribed rate terms

d
 g
dy
Problem: In general CFD codes only admit to a single value
for pressure, p. How to solve for two ‘phases’ at a given point?
Multiply-shared Space method (MUSES) developed in 1990’s
to solve this problem.
4
SOLID OXIDE FUEL CELL
Converts chemical energy to electricity and heat.
Basic components anode, cathode, electrolyte.
O2- ions produced at cathode combine with electrons and H2,
at anode to form H2O.
Operate at 800-1 000 °C,
Thermally induced stresses, and performance of cell
important.
Cells ‘stacked’ together to increase voltage; Fuel/air introduced
through manifolds. Stainless steel interconnects make
electrical connection. Uniform flow/pressure important.
5
SOLID OXIDE FUEL CELL
Many CFD companies now developing PEM and SOFC
models – None of these codes can be scaled to model stack.
Numerous flow channels. Same meshing problem as for heat
exchangers.
Authors have developed original stack model.
System treated as ‘sandwich’ of four materials; air, fuel,
electrolyte (including electrodes) and interconnect.
Idealised geometry: All fluid and solid regions simple
rectangular-shaped zones; planar ducts - Allowed for
comparison with detailed model for SOFC stack.
Single-cell and 10 cell stack considered: Fuel/air in cross flow.
6
SOLID OXIDE FUEL CELL
7
THERMODYNAMICS
Nernst potential, E, given by
0.5

x
x
RT
RT
H 2 O2 
0


EE 
ln

ln pa

2 F  xH 2 O  4 F
When current flows, actual potential, V, is lower,
V  E  i ' ' re  ha  hc  E  i ' ' r
i’’ current density (A/m2), h are ‘overpotentials’, re is electrolyte
resistance All lumped together as a internal resistance, r.
Goal to develop DRA model for SOFC stack design. Detailed
numerical model (DNM) used to validate model
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THEORY
General form:
 i  i i 

    i ui i    i  a ij  j  i      i i    i S ' ' '
t
j
(i)
(ii)
(iii)
(iv)
(v)
(i) Transient, (ii) convection, (iii) inter-phase transfer, (iv)
diffusion or within phase transfer, and (v) source.
i  i  







u
 i '''
1. Continuity:
i i i   m
t
Mass sources computed from Faraday’s law. Coded as
volumetric sources.
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THEORY
2. Momentum:

 



i  i ui  
 
2
   i  i ui ; ui   i pi  Fi i 0  ui      i  i ui 
t
F is ‘distributed resistance’. If
F
f  a Re
then .
2a 
 Dh2
Viscous terms in zero fluids, F is zero in manifolds.
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THEORY
3. Heat transfer


 i i c pi ti  

   i i uiTi   i  a ij T j  Ti     i kT   i q ' ' '
t
j
Heat sources due to Joule and Peltier effects. Two terms
combined as single volumetric term .
q ' ' '  i' ' ' E  V 
Heat transfer coefficient
aV  UA
a
b
Mass transfer effects accounted for by using

a * exp b   1
bm
 ''' a *
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THEORY
4. Mass transfer
Mass sources/sinks

 i mi  

    i u mi      i mi    i ji ' ' '
t
ji ' ' '  m ' ' ' mi ,t
‘T-state’ value.
Wall (not bulk) values are needed for Nernst equation
mb  mt B
mw 
1 B
Mass transfer driving force B  expb 1 for 1-D
convection/diffusion
 '' g *
Blowing parameter b  m
g* is conductance zero mass transfer
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MUSES method
Multiply-shared Space method. Separate spaces for each
material. Porosities used (except manifolds). Inter-phase
values fetched across structured rectilinear mesh.
13
Implementation
FORTRAN code inPLANTed in PHOENICS code:
PATCH (fu2el,VOLUME,NI1,NI2,NJ1,NJ2,NK1,NK2,1,LSTEP)
<SORC01> VAL=TEM1[,,+:FTOE:]
COVAL(fu2el,TEM1,GRND,GRND)
IF(INDVAR.EQ.INAME('TEM1 ').AND.NPATCH.EQ.'FU2EL ') THEN
LFVAL =L0F(VAL)
LFTEM1=L0F(INAME('TEM1 '))
DO 13801 IX=IXF
,IXL
IADD=NY*(IX-1)
DO 13801 IY=IYF
,IYL
I=IY+IADD
L0TEM1=LFTEM1+I
13801 F(LFVAL+I)=F(L0TEM1+NFM*42)
ENDIF
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COMPUTE ALGORTHM
1. Solve transport equations.
2. Solve Nernst potential,
3. Compute values for cell resistance, r, and heat and mass
sources/sinks.
4. Repeat steps 1-3 until convergence is obtained.
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VOLTAGE ‘CORRECTION’
Both galvanostatic and potentiostatic situations occur. For
former adjust voltage iteratively until the desired current
reached.
~
V V * V
~
V   r i ' 'i ' '*
V* is V-value of at previous iteration and V~ is a voltage
correction i ' ' is desired current density. This step arguably
redundant (can compute cell voltage from mean current
density direct)
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DETAILED NUMERICAL MODEL
(DNM)
Used for comparison. Rate eqns. not assumed in DNM. Fine
mesh. No volume averaging.
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SINGLE CELL RESULTS :
Current density
DRA
DNM
Both cell temperature and Nernst potential affect the current
density since i' '  E  V  r
r is inversely proportional to temperature
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Nernst Potential
DRA
DNM
Nernst potential more influenced by H2 and H2O mass factions
than by O2, due to the stoichiometry. E decreases as
concentrations of O2 and H2 decrease and H2O increase from
inlet to outlet.
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Temperature
DRA
DNM
Temperature variations are inevitable: If heat production
uniform, maximum would be at top-right corner.
Since both E and i’’ are maximum at top-left; peak shifts
Metallic interconnect serves as a thermal fin
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Anode wall H2 mass fraction
DRA
DNM
H2 contours nominally perpendicular to, and decreasing along
fuel streamlines
NB: DRA values wall values
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Anode wall H2O mass fraction
Good agreement between methodologies.
DRA
DNM
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Cathode wall O2 mass fraction
DRA
DNM
Current density identical at anode/cathode (thin electrolyte), so
source terms of H2, H2O and O2 coupled, and mass fraction
contours skewed, more pronounced at high current densities.
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10-CELL STACK RESULTS:
Air side velocity vectors
i ' '  4 000 A/m 2
DRA
DNM
Fine detail of motion lost,
Characteristic parabolic-shaped (DNM) velocity profile
associated with fluid flow in planar ducts absent from DRA
Manifold-stack assembly ‘well designed’ pressure/velocity
fields uniform: Manifolds losses small compared to cell.
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Air side pressure
DRA
DNM
Pressure fields in close agreement.
Problems can arise in large stacks, where mass transfer into
results in stack pressure gradient decreasing upwards. results
in variations in flow field. Minimised by ensuring cell passages
small in comparison manifolds.
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Air side bulk O2 mass fraction
DRA
DNM
O2 mass fractions constant from cell-to-cell. Good agreement
between methodologies. NB: DRA values bulk values,
Significant variation across micro-channels; maximum for high
current density (short circuit, V  0) ‘diffusion limit’; mass
transfer rate-limiting factor. Important that wall values obtained
to avoid over-prediction of Nernst potential from bulk values
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Temperature
DRA
DNM
‘Zig-zag’ temperature field in
DNM
Secondary thermal
distribution due to ‘ordering of
fluids’
Occurs even if heating
perfectly uniform and flow
constant
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DISCUSSION
Problem: In original DRA secondary thermal effects lost, due
to volume-averaging.
Solution:
Computational cells in z direction coincided with actual fuel
cells. Inter-phase heat transfer terms computed as pairs eg.
q ae ' ' '   qea ' ' '  a ae Ta k   Te k 
air-electrolyte pair, Similarly treatment for air-interconnect, fuelinterconnect but for fuel-to-electrolyte pair sources computed
as.
q fe ' ' '  a ae T f k  1  Te k 

qef ' ' '  a ae Te k  1  T f k 
Recovered ‘ordering of streams’
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DISCUSSION
Complex interaction between physical chemistry and transport
phenomena; subject known as ‘physical-chemical
hydrodynamics’
Voltage correction algorithm converged provided a reasonable
initial guess is made.
If V-i’’ curve is linear, and r is actual resistance, correct cell
value predicted after 1 iteration: In practice V-i’’ curve nonlinear and r only an estimate
Advantage voltage-correction approach, is r need not be exact,
Alternative can compute the average resistance and current
density by integration, and hence obtain voltage. Same result
is obtained.
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DISCUSSION
PLANT/IMMERSOL good for ‘in-line’ programming, but need
way to nest multi-line commands e.g. using curly braces {} or
the like.
In some aspects (eg voltage correction) PLANT is a little
clumsy. (Compromise)
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CONCLUSIONS
Most CFD vendors developing single cell models which
cannot be scaled to stack level
We developed first CFD stack-level model
Excellent agreement between DRA and DNM results
Substitution of appropriate values for a and F leads to
reasonable results at fraction of cost
1-D mass transfer analysis yields wall mass fractions in
agreement with detailed calculations even when significant
variations in mw and mb
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[email protected]
[email protected]
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